GÖDEL'S INCOMPLETENESS THEOREMS AND ARTIFICIAL LIFE

John P. Sullins III, San Jose State University

In this paper I discuss whether Gödel's incompleteness theorems have
any implications for studies in Artificial Life (AL). Since Gödel's
incompleteness theorems have been used to argue against certain mechanistic
theories of the mind, it seems natural to attempt to apply the theorems to
certain strong mechanistic arguments postulated by some AL theorists.

We find that an argument using the incompleteness theorems can not be
constructed that will block the hard AL claim, specifically in the field of
robotics. However, we will see that the beginnings of an argument casting
doubt on our ability to create living systems entirely resident in a computer
environment might be suggested by looking at the incompleteness theorems from
the point of view of Gödel's belief in mathematical realism.

1. INTRODUCTION

For many decades now it has been claimed that Gödel's two incompleteness
theorems preclude the possibility of the development of a true artificial
intelligence which could rival the human brain.[1] It is not my purpose to rehash these argument in terms of
Cognitive Science. Rather my project here is to look at the two incompleteness
theorems and apply them to the field of AL. This seems to be a reasonable
project as AL has often been compared and contrasted to AI (Sober, 1992;
Keeley, 1994); and since there is clearly an overlap between the two studies,
criticisms of one might apply to the other. We must also keep in mind that not
all criticisms of AI can be automatically applied to AL; the two fields of
study may be similar but they are not the same (Keeley, 1994).

Gödel himself realized that the incompleteness theorems alone do not
preclude the possibility of a machine mind (Wang, 1987, pg. 197). In fact
there is an interesting argument posed by Rudy Rucker where he shows that it is
possible to construct a Lucas style argument using the incompleteness theorems
which actually suggests the possibility of creating machine minds (Rucker,
1983, pp. 315-317). Arguments like Rucker's point out the inadequacy of using
the incompleteness theorems alone to try to prove the improbability of machine
minds. In fact an important part of understanding Gödel's reluctance to
accept the project of AI stems from his belief in mathematical realism (see
Tieszen, 1994, for a full discussion of this point). My purpose here is not to
try to convince anyone of the validity of the Penrose-Lucas arguments in
cognitive science, but rather to see how a similar argument might be applied to
the field of AL. I will endeavor to keep my arguments as close to those that
Gödel himself might have made if he had been presented with the ideas
expressed in strong AL. It may turn out that the incompleteness theorems have
no relevance to AL, but we must take a closer look before we dismiss them out
of hand.

AL does not start out with the goal of modeling human intelligence; rather it
is interested in studying life at a fundamental level, comparing and
contrasting our knowledge of "life-as-we-know-it within the larger
picture of life-as-it-could-be" (Langton, 1987, pp. 1). AL begins with
modest goals. Examples of AL projects would range from the modeling of actual
biological processes like the life cycles of slime mold (Resnick, 1994, p. 50),
to the creation of simple "artificial ecosystems" like Thomas Ray's Tierra
program, a system entirely resident in a computer which makes few claims to
be an accurate representations of life-as-it-is while still claiming to be
some new form of synthetic life (Ray, 1992, p. 371). So we can see that at
least some of the researchers in the field of AL do claim that these creations
are (or could be) in a real sense an actual member of the set of things living
(see Emmeche, 1994, p. 3).

There are clearly two ways to approach AL models: one is to consider them
tools for studying the natural world, and the other is to claim that AL
programs, properly executed, simply are living things (Sober, 1992, p. 749).
It seems to me that Gödel's theorems will have little impact on the former
claim as it already concedes that AL is simply a modeling technique for and not
an instantiation of life. Conversely, though, Gödel's theorems probably
do apply to the much stronger latter claim that AL can currently, or will
eventually, create artificial living things. This is because the later claim
suggests that an artificially constructed reality can completely capture the
minimum necessary criteria for the creation of life and, as we will see later,
Gödel's theorems can be argued to imply that this may be problematic.

2. GÖDEL'S VIEWS ON MECHANISM IN BIOLOGY

I was spurred in the direction of applying Gödel's theorems to AL when I
came upon the following passage in Hao Wang's From Mathematics to
Philosophy, where he is discussing Gödel's views on the relationship
between minds and machines:

Gödel believes that mechanism in biology is a prejudice of our time which
will be disproved. In this case one disproval, in Gödel's opinion, will
consist in a mathematical theorem to the effect that the formation within
geological times of a human body by the laws of physics (or any other laws of a
similar nature), starting from a random distribution of elementary particles
and the field, is about as unlikely as the separation by chance of the
atmosphere into its components.

Mechanistic or closely related reductionistic theories have been part of
theoretical biology in one form or another at least since Descartes. I do not
want to give the impression that I believe that mechanistic or reductionistic
theories form some kind of monolithic doctrine. I realize that there are
probably as many different versions of these arguments as there are theorists
in the field of biology. Later in this paper I will specify which brand of
mechanism and reductionism is employed in strong AL arguments.

The various mechanistic and reductionistic theories are historically opposed
to the much older and mostly debunked theories of vitalism (see Emmeche, 1991).
These theories (the former more than the latter), along with formism,
contextualism, organicism, and a number of other "isms" mark the major centers
of thought in the modern theoretical biology debate (see Sattler, 1986).

It occurs to me that AL falls curiously on many sides of these debates in the
philosophy of biology. For instance AL uses the tools of complete
mechanization, namely the computer, while at the same time it acknowledges the
existence of emergent phenomena (Langton, 1987, p. 81). Neither mechanism nor
reductionism is usually thought to be persuaded by arguments appealing to
emergence. Facts like this should make our discussion interesting. It may
turn out that AL is hopelessly contradictory on this point, or it may provide
an escape route for AL if we find that Gödel's incompleteness theorems do
pose a theoretical road block to the mechanistic-reductionistic theories in
biology which I will outline later.

What I will attempt to do now is to take a look philosophically at how AL
relates to a specific form of the mechanistic and reductionistic philosophies
of biology and then apply Gödel's incompleteness theorems to that specific
view in an attempt to determine if the project of AL can avoid the problems
experienced by AI in its encounters with Gödel.

3. MECHANISM AND REDUCTIONISM IN BIOLOGY

In this paper I will be discussing only two of the above mentioned world
views, namely, mechanism and its closely related theory reductionism.
Furthermore, I will be concerned only with specific formulations of mechanistic
and reductionistic theories. This means that we need to be very clear in
describing just what we mean by the terms "mechanism" and "reductionism," as
they are often used in many different contexts and their meanings can change
subtly depending on their use. After we have an adequate understanding of the
basic assumptions found in the various mechanistic and reductionistic
philosophies of biology, we can then determine if the underlying metaphysical
assumptions in AL theories should be placed under this heading.

The history of the idea of mechanism is an interesting one, but I will not
retell it here. We should understand, however, that it received its greatest
boost in popularity in the seventeenth century as a reaction to the new science
of physics on the part of those studying natural philosophy. As we all know,
the capacity of physics to explain, model, and predict things like planetary
orbits astounded the scientific community in the seventeenth century. It
occurred to many thinkers of that time that many biological things might also
be explainable, modelable, and predictable using the basic laws of physics as
they relate to machinery. After all, if one looks at a body it does seem to be
a machine of some sort with, for instance, lever actions explaining the
workings of muscles and limbs among other things. Descartes was willing to
describe all animals as simple machines; possibly even the human body could be
reduced this way. But he was not willing to so describe the human mind.
Bolder thinkers such as De la Mettrie (1748) were willing to push the metaphor
to the limits, describing humans completely as machines. The metaphor of the
machine or "clockwork body" is still prevalent today. In this period of rapid
discoveries in physics and mechanics we find wonderful early AL experiments
consisting of clockwork people and animals which were built as objects of
amusement in the seventeenth and eighteenth centuries (see Emmeche, 1991, and
Langton, 1987).

Over the centuries, the mechanistic and the closely related reductionistic
theories of biology have keep pace with current discoveries in science until
today a mechanist can be thought of as one who believes "that an organism is in
reality nothing more than a collection of atoms, a simple machine made of
organic molecules" (Emmeche, 1991, p. 12). We should note that mechanism, like
all theories, changes over time. To be fair, we should realize that the
mechanistic theories in biology that Gödel would have been referring to
(in the quote above) have changed and are slightly different today. In the
late sixties one could have found many mechanistically inclined theorists who
would claim that it was self-evident that, since biological entities are
physical they must obey the laws of mechanics, and that meant that living
systems were simply matter in motion obeying the laws of classical mechanics
(Sattler, 1986, p. 216). But physics has gone far beyond classical mechanics,
and many biological mechanists would now agree that it is not possible to
accurately describe a living system using only classical mechanics (Sattler,
1986, p. 216). This is perfectly reasonable. If it is generally accepted
that classical mechanics is unsuitable for a complete understanding of
nonliving matter, then how can it be expected to be sufficient for explaining
the much more complex actions of living matter (Sattler, 1986)? So it is safe
to say that most theorists have outgrown the idea that life can be explained
wholly in terms of classical mechanics. Instead, what is usually meant is the
following (paraphrased from Sattler, 1986):

1) Living systems can and/or should be viewed as physico- chemical systems.

2) Living systems can and/or should be viewed as machines. (This kind of
mechanism is also known as the machine theory of life.)

3) Living systems can be formally described. There are natural laws which
fully describe living systems.

Now it is not necessary for one to hold all three of the above statements in
order to be a biological mechanist. All one has to do is believe at least one
of the above statements. So a mechanist believes, basically, that living
systems can be completely explained by the operation of the physical
laws of matter, such as classical mechanics, quantum mechanics, complexity
theory, etc. Any particular mechanist may think that we do not yet have within
our grasp all of the laws we need to understand life, but no mechanist will say
that we cannot theoretically discover them in a reasonable amount of time.

Reductionism is related to mechanism in biology in that mechanists wish to
reduce living systems to a mechanical description. Reductionism is also
the name of a more general world view or scientific strategy. In this world
view we explain phenomena around us by reducing them to their most basic and
simple parts. Once we have an understanding of the components, it is then
thought that we have an understanding of the whole. There are many types of
reductionist strategies. To help clarify the different categories of
reductionism I will turn to the work of John R. Searle. Searle lists five
different reductionist strategies in his book, The Rediscovery of the
Mind. These are Ontological Reduction, Property Ontological Reduction,
Theoretical Reduction, Logical or Definitional Reduction, and Causal Reduction
(Searle, 1992). And to this list we should also add Epistemological and
Methodological Reduction (see Bonabeau and Theraulaz, 1994, and Sattler, 1986).
This complexity causes much confusion when one tries to discuss the concept of
reductionism, so we should briefly describe each of these strategies.

Ontological reductionism in theoretical biology occurs when a theory states
that a living system is nothing but a collection of physical parts (atoms)
being acted upon by the laws of physics. This can be abstracted further by
saying that the laws of physics are nothing but a set of formalizable axioms
which can be understood separate from physical matter. "Hence, a complete
knowledge of the physics and chemistry of life would entail a full
understanding of life" (Sattler, 1986, p. 218). This concept applies to AL
theories that promote the belief that, "Since we know that it is possible to
abstract the logical form of a machine from its physical hardware, it is
natural to ask whether it is possible to abstract the logical [form] of an
organism from its biochemical wetware" (Langton, 1987, p. 21).

Property ontological reduction can occur in theoretical biology and in AL when
one attempts to describe a property or behavior of a living thing by appealing
to low-level phenomena or rules which dictate the behavior. An example of
property ontological reduction in AL would be if some one claimed that the
flocking behavior of birds could be completely reduced, for instance, to the
workings of Craig Reynolds's famous boids program.[2]

Theoretical, or, as it is sometimes called, epistemological, reductionism is
the belief that the theories of one science can be reduced to the theories of
another. "In biology the central question of epistemological (theoretical,
explanatory) reductionism is whether the laws and theories of biology can be
shown to be special cases of the laws and theories of the physical sciences"
(Dobzhanaky, et al., 1977, p. 491, as quoted in Sattler, 1986, p. 221).
In AL this brand of reductionism appears when the claim is made that the laws
of nature might be reducible or capturable in the laws surrounding the
information processing of computation.

Logical or definitional reductionism "is a relation between words and
sentences, where words and sentences referring to one type of entity can be
translated without any residue into those referring to another type of entity"
(Searle, 1994, p. 114). This occurs in AL when we use terms usually used in
biology to describe events that occur in our computer simulations, not
metaphorically but descriptively. For instance, the words "population,"
"organism," "fitness," etc., are all used interchangeably in AL when describing
real and artificial life forms.

Causal reductionism "is a relation between any two types of things that can
have causal powers, where the existence and a fortiori the causal powers of the
reduced entity are shown to be entirely explainable in terms of the causal
powers of the reducing phenomena" (Searle, 1994, p. 114). This seems to occur
in biology when one describes phenotype as being nothing but the actualization
of the genotype. And this occurs in AL when we say, unarguably, that the
observed behavior of a program is nothing more than the implementation of its
program code.

Finally, methodological reductionism in biology is the claim that living
systems should be studied at their most basic level, either the actual atoms
and molecules or their theoretical interactions (Sattler, 1986, p. 224).
Clearly this occurs in AL when it is suggested that we can gain understanding
of the real world by seeing it as the interaction of "information" at either
the cellular level or at the level of the patterned interaction of electrons in
circuit boards (see Rucker, 1987, for an example).

So we can see that reductionism is a tool or strategy for solving complex
problems. There does not seem to be any reason that one has to be a mechanist
to use these tools. For instance one could imagine a causal reductionistic
vitalist who would believe that life is reducible to the elan vital or
some other vital essence. And, conversely, one could imagine a mechanist who
might believe that living systems can be described metaphorically as machines
but that life was not reducible to being only a property of mechanics.

4. MECHANISM AND REDUCTIONISM IN STRONG AL

As this paper is concerned with strong AL arguments, I will narrow down our
discussion of the various reductionistic and mechanistic theories of biology to
the specific types commonly found in strong AL claims. The strong argument
claims that AL simulations are, or can be, complete in their formalization of
the basic laws describing living systems.

Now since Gödel's incompleteness theorems apply specifically to systems
which attempt to completely and consistently axiomatize arithmetic, and
generally only to systems which attempt to completely and consistently
axiomatize their subject (Nagle and Newman, 1958, p. 100, Braithwaite, 1962, p.
1). So If we refer to the three mechanistic theories of life listed above we
can begin eliminating the ones that do not apply to the strong AL conception of
living systems. With this in mind we can eliminate number 1 from the list
above, as the strong variety of AL does not believe that living systems should
only be viewed as physico-chemical systems. AL is life-as-it-could-be,
not life-as-we-know-it (Langton, 1989, p. 1), and this statement
suggests that AL is not overly concerned with modeling only physico-chemical
systems. Postulates 2 and 3 seem to hold, though, as strong AL theories
clearly state that the machine, or formal, theory of life is valid and that
simple laws underlie the complex, nonlinear behavior of living systems
(Langton, 1989, p. 2).

As far as reductionism is concerned, AL theories taken as a whole clearly fit
into all the above categories of reductionism (for some discussion of this
point see Bonabeau and Theraulaz, 1994, p. 314). But the strong claim in AL
clearly relies heavily on property reductionism, causal reductionism, and
methodological reductionism, so we can remove the other types of reductionism
from our discussion.

Having clarified what we mean by the terms mechanisim and reductionism, we can
now formulate a concise statement of the general beliefs of strong AL theories
as follows:

1. Living systems are properly reducible to the laws described in the
theories of complex adaptive systems.

2. Since a complex adaptive system is causally and methodologically reducible
to the mechanistic processes involved in the computation of
information at the fundamental level in nature, it is then conceivable
that one could completely formalize all of the laws operating in such a
system.

3. These laws can be implemented on the proper type of computing machinery.

Conclusion: A properly conceived AL program running in a complex enough
computer or robot can correctly be said to be alive.

Now that we have a clearly stated expression of the strong AL claim, we are at
the point where we can apply Gödel's incompleteness theorems to the
argument. I believe that Gödel's incompleteness theorems have some
bearing on the question of the validity of the strong claim in AL since the
second premise just listed makes a claim to a level of formal completeness that
may be subject to the limitations of formal systems described by Gödel.

5. GÖDEL'S INCOMPLETENESS THEOREMS APPLIED TO AL

In order to show that Gödel's incompleteness theorems have a bearing on
AL, we have to prove that it is necessary for strong AL to hold to postulate
number 2 as I have stated it above. In order to achieve this I will use Steen
Rasmussen's (1992) article, "Aspects of Information, Life, Reality, and
Physics" (p. 767), as it does a wonderful job of laying out the logical steps
taken in the strong AL argument. Briefly stated, his argument goes like
this:

2. Life is a physical process. Corollary: 1, Hence life can be
simulated on a universal computer.

3. There exist criteria by which we are able to distinguish living from
non-living objects. Corollary 2: From this postulate it follows that it is
possible to determine if some specific computer process is alive or not.

4. An artificial organism must perceive a reality R2 , which, for
it, is just as real as our "real" reality, R1 , is for us (R1
and R2 may be the same).

5. R1 and R2 have the same ontological status. Using
postulate 5 and Corollary 1 we can say that the ontological status of a living
process is independent of the hardware that carries it. Since R1 and R2 are ontologically equal, that is, one
is not more real than the other, then actual living systems can be created in
a digital computer.

6. It is possible to learn something about the fundamental properties of
realities in general, and R1 in particular, by studying the details
of different R2's. An example of such a property is the physics of
a reality.

Postulates 1, 2, and 3 are not completely unproblematic but I will
not take that up here; rather we will jump to postulates 4 and 5. In postulate
4 Rasmussen rightly claims that in order for an AL program to be alive it has
to create an environment that is as real to its inhabitants as nature is to us.
In explaining this idea he appeals to a concept called a

"Meaning Circuit." The basic idea behind this concept is that the world is a
self-synthesized system of existence. On the one hand, physics provides the
means for communication (light, sound, etc.). Reality can, thereby, acquire
its meaning through a conscious conception of the world, via an organization of
the information we get from our senses. On the other hand, physics also gives
rise to chemistry and biology, and through them, an observer participation,
namely the emergence of life and later the evolution of man (Rasmussen, 1992,
p. 769).

So what postulate 4 is saying is that the living systems in an artificial
reality must have some form of robust interaction and awareness of that
reality and this interaction, this "meaning circuit," is what makes the
artificial reality real. In postulate 5 an interesting jump is made.
He claims that, "In postulate 4 we argued that a reality obtains its meaning
through the existence of an observer" (Rasmussen, 1992, p. 770). He then goes
on to explain that the artificial reality is a real reality whenever it has a
living agent interacting with it. If this is achieved then R1 and
R2 have equal ontological status (Rasmussen, 1992, p. 770).

The problem with this argument so far is that it seems to be circular. It is
making the claim that an artificial reality created in the computer is able to
capture all of the essential qualities of our reality (R1is equal to
R2) as long as living agents are interacting with the system, but
the artificial reality must already be ontologically equivalent to our reality
in order to produce truly living artificial life forms. So in a sense the
argument is saying that in order to create artificial life one needs to have
artificial life to create the proper artificial environment with the right
ontological status. Which comes first? I believe that this is a serious flaw
in the strong AL argument, and it may be much more difficult to get around than
any of the arguments which will be posed below.

Let us assume that we can get around the circularity of the argument just
described. According to postulate 4, the artificial reality experienced by the
artificial life agent must be as real to it as our reality is for us. Using
the concept of the meaning circuit as described above, it is necessary, in
order to capture the essential qualities of the reality we perceive, for an AL
program to have some form of internal logic equivalent to the physics we
perceive in nature so as to provide the artificial organisms with the same kind
of meaningful interaction with their world which organisms in our reality
experience. This physics can be a simplified version of the one we experience
in our reality (Rasmussen, 1992, p. 769), but it must be a complete
formalization of a certain number of basic physical laws required for the
existence of life. For instance, there must be some way for the agents and the
environment to interact. Since we are programming a computer to invoke this
environment, then this set of basic physical laws must be one that can be
formed into specific statements in which the program will mechanically deduce
the environment and the agents in that environment. We can state this as a
postulate:

There exists a minimum set of formal axioms which can be used to create a
complete artificial physics capable of sustaining artificial life.

Now here is the tricky part. One of the main differences between an actual
living organism and its potential AL counterpart is that the AL entity exists
in a computer. Also a living creature is presented with the physics of the
natural world, where an AL entity has to have its physics provided by the
computer. So in accord with the above postulate, a programmer must code into a
computer system the minimum set of formal axioms needed to create a complete
artificial physics capable of sustaining artificial life. In order to become a
proper artificial physics capable of sustaining life the program used would
have to be able to simulate a reality that is as real to its inhabitants as
ours is to us. Now if we hold to a level of mathematical reality as strictly
as Gödel does, then concepts like arithmetic are as real an entity as
anything else we experience; specifically, a mathematical realist like
Gödel believes that our intuitions, expressed by mathematics, are about,
"abstract, mind-independent meanings and objects, including transfinite
objects" (Tieszen , 1994). As we know, Gödel's incompleteness theorems
seem to have proven that building a consistent formalized system of proving all
arithmetic truths is highly unlikely (Gödel, 1962, p. 77, Nagle and
Newman, 1958, p. 99). Simply put (if that is possible), Gödel's
incompleteness theorems suggest that there exist sentences which can be
formulated in a specific formal system called Peano-Arithmetic which are true
but nonetheless not deducible from the axioms of that system. It follows from
this that it is unlikely that we currently have a complete formal system which
can grasp the entirety of even simple mathematical systems. This means (as
long as you are a mathematical realist) that at least one of the basic
qualities of our reality will always be missing from any conceivable
artificial reality, namely, a complete formal system of mathematics. This
argument tends to make more sense when applied to strong AI claims about
intelligent systems understanding concepts (see Tieszen, 1994, for a more
complete argument as it concerns AI).

Still, I feel that it has relevance to AL for two reasons. The first is that
even though the intelligence of a typical postulated AL entity is small, it is
hoped that greater intelligences will evolve in time from these modest roots.
So, if we are to believe that AL can eventually evolve higher intelligences,
we need to know how it can avoid the typical arguments deployed against strong
AI claims such as the Gödel argument. Secondly, while one might also ask
what possible effect these postulated mathematical realities have on living
systems, real or artificial, I believe that it can be argued that some form of
mathematical realism is not unthinkable and that this condition of our reality,
coupled with Gödel's theorems, casts doubt on our ability to render an
artificial reality which would be equal to our own reality in its ability to
sustain life. To illustrate this idea let us look briefly at a quote from John
von Neumann regarding mathematics and AI:

When we talk mathematics, we may be discussing a secondary language,
built on the primary language truly used by the central nervous system.
Thus the outward forms of our mathematics are not absolutely relevant
from the point of view of evaluating what the mathematical or logical language
truly used by the central nervous system is (quoted by Weizenbaum,
1976).

It seems that one could broaden the scope of von Neumann's observation from
the specifics of a living central nervous system to life in general without
harming the intent of the original comment. I feel that this is the position
that a mathematical realist like Gödel would take because a mathematical
realist would believe that there exist mathematical realities which are the
foundations of the reality we experience and that these realities are described
by concepts like Peano-Arithmetic, but that these realities are uncapturable in
any complete way by entirely mechanical processes. Thus it would seem that it
is impossible to completely formalize an artificial reality that is
equal to the one we experience, so AL systems entirely resident in a computer
must remain, for anyone persuaded by the mathematical realism posited by
Gödel, a science which can only be capable of potentially creating
extremely robust simulations of living systems but never one that can
become a complete instantiation of a living system.

6. OBJECTIONS

The argument that I have presented above is admittedly brief. In a
short paper such as this it is hard to adequately defend a theory that makes
use of Gödel's theorems as seen from the perspective of his mathematical
realism. Both of these subjects would take up the better part of a book to
thoroughly explain. My purpose here is only to open a discussion of this topic
in the hope that others agree that it is a worthwhile subject for further
study. In fact I hope to collect many objections to the argument so that I can
attempt to answer them later in a more thorough way.

Still it would be helpful here to look at the most common objection that I
have received to this argument and attempt to begin a counter argument.

Those to whom I have shown earlier drafts of this paper usually point out an
objection similar to this. Our reality (R1 ) is a reality in which
the incompleteness theorems hold. So why does it matter that the
incompleteness theorems hold in an artificial reality (R2)? All the
above argument has accomplished is to point out that Gödel's theorems are
valid in both R1 and R2. Also, computers already do some
amazing things none of which requires the strict formal completeness and
consistency that Gödel is worried about in his famous theorems.

It is true that the incompleteness theorems hold to our perceived reality and
that they point to a fundamental limit in our ability to formalize all of our
mathematical intuitions. I do not believe that Gödel meant to suggest
that mathematics as a separate entity is fundamentally incomplete. Rather, his
theorems prove that our understanding of that mental object known as
mathematics can not be completely and consistently mechanized. So what I am
saying is this: given Gödel's mathematical realism, the incompleteness
theorems suggest that it is not possible to capture this one aspect of our
reality in any artificial reality. If one assumes that our universe is
infinite,"then it embodies the full set of natural numbers, so Gödel's
theorem seems to say that for any given finite theory of the universe, there
are certain facts having to do with sets of physical objects that can not be
proved by the theory" (Rucker, 1982, p. 141). Now any AL program that is
attempting to entirely create an environment separate from our own which is
capable of sustaining life is attempting to capture the sufficient conditions
which make life possible here. I am claiming that Gödel's theorem
suggests that any such program might be missing an important essential portion
of our reality, namely, its fundamental mathematical reality, so that the
artificial reality (R2) would not be ontologically equal to our
reality (R1). And since this is a requirement for creating truly
living artificial life entities, the artificial reality could not sustain life.

7. SO WHAT

Now I will try to mitigate some of the consequences of the above argument and
suggest ways that AL can avoid the argument or change to accommodate it.

We should not feel that AL is diminished if it proves to be impossible to
synthesize living systems in the manner described above. AL in its so called
"weak" form is still a challenging new science which promises to completely
alter the way we practice the study of biology by giving us powerful new tools
and metaphors for looking at and discussing living systems (Emmeche, 1994, p.
156). Secondly, the argument given above only applies to AL experiments
completely carried out within a computer.

When we look at the argument above we can see that all it suggests is that
there is not a complete one-to-one correspondence between nature and a
simulated nature. Remember that the artificial organism must perceive a
reality that is as real to it as our reality is to us (Rasmussen, 1992, p.
769). Since there may be some problem with a simulated reality, then that
problem can be solved by allowing the artificial organism to interact with our
reality. This can be done through robotics.

In this scheme the robotic artificial organisms are operating in an unarguably
real environment. If a way could be found to give the robots complex adaptive
behavior and self reproduction then we might be on our way to creating true
artificial life. It may be possible, but certainly not easy, to evolve living
organisms from robots.

8. CONCLUSION

We have seen that due to a specific interpretation of the implications of
Gödel's incompleteness theorems it may not be possible to create a truly
living system which is entirely resident in a computer. We were not able to
advance very far Gödel's claim that mechanism in biology can be disproven
mathematically. We have only proven that life may not be reducible to a
certain type of mechanical implementation on a computer. This modest result
may lead to a more complete refutation of mechanism, but that question is left
open for now. It may be that studies in AL itself will lead to the
mathematical proof that Gödel postulated in the quote above.

The value of this finding is not to discourage certain types of research in
AL, but rather to help move us in a direction where we can more clearly define
the results of that research. In fact, since one of the above arguments rests
on the assumption that the universe is infinite and that some form of
mathematical realism is true, if we are someday able to complete the goal
advanced in strong AL it would seem to cast doubt on the validity of the
assumptions made above. So succeed or fail AL gives us much to ponder.

It may be that AL is still a long way from capturing completely the answer to
the question "what is life?" It may be that this question is unanswerable or
the wrong question to ask. But every attempt at answering that question, from
modest attempts in AL at the explication of life, to extreme attempts in strong
AL to synthesize life, helps us move closer to an understanding of the world we
find ourselves in.

ACKNOWLEDGMENT

I would like to thank the staff and faculty of the San Jose State University
philosophy department for their support of my studies. I am also indebted to
Dr. S. D. N. Cook for his critique and support of this project, Dr. R. Rucker
for his scathing criticisms, and Dr. R. Tieszen for his comments on earlier
drafts of this paper. This work has been partially supported through a grant
from the National Science Foundation.